Finite Group Databasebeta

← #22586342

Subgroups — #22586342

Order 9088.

168 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17985916C211yesmaximal, characteristic
2#17985725C211yesmaximal, characteristic, complemented
3#343×C71C211yesmaximal, characteristic, complemented
4C22×D284C411yescharacteristic
5C2×(C2×D71)⋊C4—21no—
6C2×(C2×D71)⋊C4—21no—
7C22⋊(C71⋊C8)—21no—
8C71⋊#95C411yescharacteristic
9C2×M16×C71—21no—
10C2×C4⋊C4×C71C2211yescharacteristic
11C23×D71—21no—
12C2×D284—41no—
13C2×D284—42no—
14C22×Dic71—21no—
15C2×C71⋊C8—41no—
16(C2×D71)⋊C4—42no—
17(C2×D71)⋊C4—41no—
18(C2×D71)⋊C4—41no—
19C22×C4×C71C2×C411yescharacteristic
20C22×C4×C71D411yescharacteristic
21C22×C4×C71D411yescharacteristic
22M16×C71—22no—
23M16×C71—41no—
24C2×C8×C71—41no—
25C4⋊C4×C71—41no—
26C23×C71C22⋊C411yescharacteristic
27C22×D71—22no—
28C22×D71—41no—
29C22×D71—41no—
30C22×D71—41no—
31C22×D71—41no—
32C22×D71—41no—
33C22×D71—41no—
34C2×C4×C71—21no—
35C2×C4×C71—21no—
36C2×C4×C71—21no—
37C2×C4×C71—21no—
38C2×C4×C71—41no—
39C2×C4×C71—41no—
40C2×C4×C71D811yescharacteristic
41C2×C4×C71SD1611yescharacteristic
42C2×Dic71—41no—
43C2×Dic71—41no—
44C2×Dic71—41no—
45D284—42no—
46D284—42no—
47C8×C71—42no—
48C71⋊C8—81no—
49C22×C71—21no—
50C22×C71—21no—
51C22×C71#9911yescharacteristic
52C22×C71#10211yescharacteristic
53C22×C71#10411yescharacteristic
54C2×D71—42no—
55C2×D71—42no—
56C2×D71—42no—
57C2×D71—81no—
58C2×D71—81no—
59C2×D71—81no—
60C2×D71—81no—
61C4×C71—21no—
62C4×C71—21no—
63C4×C71—41no—
64C4×C71—41no—
65Dic71—81no—
66C2×C71—21no—
67C2×C71—21no—
68C2×C71#32711yescharacteristic
69C2×C71AΣL(1,𝔽16)11yescharacteristic
70C2×C71#35511yescharacteristic
71D71—82no—
72#1345—711nomaximal, Sylow
73C71#134511yescharacteristic, complemented, Sylow
74#394—711no—
75#343D7111yescharacteristic, complemented
76#324—711no—
77C22×D4—711no—
78C2×M16—21no—
79C2×C4⋊C4C2×D7111yescharacteristic
80C2×C22⋊C4—1421no—
81C2×C22⋊C4—1421no—
82C22⋊C8—1421no—
83#95—711no—
84C24—1421no—
85C2×D4—2841no—
86C2×D4—2842no—
87C22×C4—1421no—
88C22×C4C4×D7111yescharacteristic
89C22×C4D28411yescharacteristic
90C22×C4C22⋊D7111yescharacteristic
91M16—22no—
92M16—41no—
93C2×C8—41no—
94C2×C8—2841no—
95C4⋊C4—41no—
96C22⋊C4—2842no—
97C22⋊C4—2841no—
98C22⋊C4—2841no—
99C23—2841no—
100C23—2841no—
101C23—2841no—
102C23—2841no—
103C23—2841no—
104C23—2841no—
105C23—1422no—
106C23(C2×D71)⋊C411yescharacteristic
107D4—2842no—
108D4—2842no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—2841no—
116C2×C4—2841no—
117C2×C4—2841no—
118C2×C4#1176585011yescharacteristic
119C2×C4#1176585211yescharacteristic
120C8—42no—
121C8—5681no—
122C22—21no—
123C22—21no—
124C22—2842no—
125C22—2842no—
126C22—2842no—
127C22—5681no—
128C22—5681no—
129C22—5681no—
130C22—5681no—
131C22#1455604211yescharacteristic
132C22#1455604811yescharacteristic
133C22#1455605611yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—5681no—
139C2—21no—
140C2—21no—
141C2—5682no—
142C2#1798573211yescharacteristic
143C2#1798578711yescharacteristic
144C2#1798579511yescharacteristic
subgroupG/H
centerC22#14556042
commutatorC22×C4×C71C2×C4
FrattiniC2×C4⋊C4C2×D71
Fitting#343×C71C2
socleC22×C71#99
hypercenter#343D71
hypocenterC71#1345
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #343

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #17985725C2semidirect
3. #343×C71C2semidirect
73. C71#1345semidirect
75. #343D71semidirect